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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj873 | Structured version Visualization version GIF version | ||
| Description: Technical lemma for bnj69 35024. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj873.4 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
| bnj873.7 | ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) |
| bnj873.8 | ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) |
| Ref | Expression |
|---|---|
| bnj873 | ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj873.4 | . 2 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑔∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
| 3 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑓𝐷 | |
| 4 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑓 𝑔 Fn 𝑛 | |
| 5 | bnj873.7 | . . . . . 6 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) | |
| 6 | nfsbc1v 3808 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜑 | |
| 7 | 5, 6 | nfxfr 1853 | . . . . 5 ⊢ Ⅎ𝑓𝜑′ |
| 8 | bnj873.8 | . . . . . 6 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) | |
| 9 | nfsbc1v 3808 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜓 | |
| 10 | 8, 9 | nfxfr 1853 | . . . . 5 ⊢ Ⅎ𝑓𝜓′ |
| 11 | 4, 7, 10 | nf3an 1901 | . . . 4 ⊢ Ⅎ𝑓(𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
| 12 | 3, 11 | nfrexw 3313 | . . 3 ⊢ Ⅎ𝑓∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
| 13 | fneq1 6659 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)) | |
| 14 | sbceq1a 3799 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ [𝑔 / 𝑓]𝜑)) | |
| 15 | 14, 5 | bitr4di 289 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ 𝜑′)) |
| 16 | sbceq1a 3799 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ [𝑔 / 𝑓]𝜓)) | |
| 17 | 16, 8 | bitr4di 289 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ 𝜓′)) |
| 18 | 13, 15, 17 | 3anbi123d 1438 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
| 19 | 18 | rexbidv 3179 | . . 3 ⊢ (𝑓 = 𝑔 → (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
| 20 | 2, 12, 19 | cbvabw 2813 | . 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
| 21 | 1, 20 | eqtri 2765 | 1 ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ w3a 1087 = wceq 1540 {cab 2714 ∃wrex 3070 [wsbc 3788 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: bnj849 34939 bnj893 34942 |
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